Method of transmitting synchronization signal in wireless communication system

ABSTRACT

A method of transmitting a synchronization signal includes generating a sequence P(k) for a synchronization signal from a Zadoff-Chu (ZC) sequence having the odd numbered length N, the sequence P(k) having the even numbered length N−I, mapping the sequence P(k) to subcarriers so that the sequence P(k) is halved with respect to a DC subcarrier, and transmitting the a synchronization signal in the subcarriers. Time/frequency ambiguity caused by a synchronization error can be avoided, and sequence detection errors can be decreased.

This application claims priority to International Application No.PCT/KR2008/002763, filed on May 16, 2008, which claims priority toKorean Patent Application Nos. 10-2007-0048353, filed on May 17, 2007,10-2007-0057531, filed on Jun. 12, 2007 and 10-2007-0068364, filed onJul. 6, 2007, all of which are incorporated by reference for allpurposes as if fully set forth herein.

TECHNICAL FIELD

The present invention relates to wireless communications, and moreparticularly, to a method of transmitting a synchronization signal in awireless communication system.

BACKGROUND ART

A wide code division multiple access (WCDMA) system of the 3-rdgeneration partnership project (3GPP) uses a total of 512 longpseudo-noise (PN) scrambling codes in order to identify base stations(BSs). As a scrambling code of a downlink channel, each BS uses adifferent long PN scrambling code.

When a user equipment (UE) is turned on, the UE performs systemsynchronization of an initial cell and acquires a long PN scramblingcode identifier (ID) of the initial cell. Such a process is referred toas cell search. The initial cell is determined according to a locationof the UE at a time when the UE is turned on. In general, the initialcell indicates a cell having strongest signal level which is measured bya downlink signal.

To facilitate the cell search, a WCDMA system divides 512 long PNscrambling codes into 64 code groups, and uses a downlink channelincluding a primary synchronization channel (P-SCH) and a secondarysynchronization channel (S-SCH). The P-SCH is used to acquire slotsynchronization. The S-SCH is used to acquire frame synchronization anda scrambling code group.

In general, cell search is classified into initial cell search, which isinitially performed when the UE is turned on, and non-initial searchwhich performs handover or neighbor cell measurement.

In the WCDMA system, the initial cell search is accomplished in threesteps. In the first step, the UE acquires slot synchronization by usinga primary synchronization signal (PSS) on the P-SCH. In the WCDMAsystem, a frame includes 15 slots, and each BS transmits the PSS in theframe. Herein, the same PSS is used for the 15 slots, and all BSs usethe same PSS. The UE acquires the slot synchronization by using amatched filter suitable for the PSS. In the second step, a long PNscrambling code group and frame synchronization are acquired by using asecondary synchronization code (SSS) on the S-SCH. In the third step, byusing a common pilot channel code correlator on the basis of the framesynchronization and the long PN scrambling code group, the UE detects along PN scrambling code ID corresponding to a long PN scrambling codeused by the initial cell. That is, since 8 long PN scrambling codes aremapped to one long PN scrambling code group, the UE computes correlationvalues of all of the 8 long PN scrambling codes belonging to a codegroup of the UE. On the basis of the computation result, the UE detectsthe long PN scrambling code ID of the initial cell.

Since the WCDMA system is an asynchronous system, only one PSS is usedin the P-SCH. However, considering that a next generation wirelesscommunication system has to support both synchronous and asynchronousmodes, there is a need for using a plurality of PSSs.

An orthogonal frequency division multiplexing (OFDM) system capable ofreducing an inter-symbol interference effect with low complexity istaken into consideration as a new system for replacing the existingWCDMA. In OFDM, data symbols, which are serially input, are convertedinto N parallel data symbols. Then, the data symbols are transmittedthrough N subcarriers. The subcarrier has orthogonality in frequencydomain and experiences independent frequency selective fading. Anorthogonal frequency division multiple access (OFDMA) scheme is anOFDM-based multiple access scheme.

An OFDM/OFDMA system is feasible to a synchronization error such as afrequency offset or a time offset. Moreover, since the PSS is a firstdetected signal in a condition whether the synchronization error exists,detection performance needs to be ensured. If the PSS detection is notachieved, synchronization is not attained, resulting in delay of networkaccess.

Therefore, there is a need for a method capable of ensuring PSSdetection performance according to radio resources allocated to theP-SCH.

DISCLOSURE OF INVENTION Technical Problem

A method is sought for transmitting a synchronization signal robust to asynchronization error.

Technical Solution

In an aspect, a method of transmitting a synchronization signal in awireless communication system includes generating a sequence P(k) for asynchronization signal from a Zadoff-Chu (ZC) sequence having the oddnumbered length N, the sequence P(k) having the even numbered lengthN−1, mapping the sequence P(k) to subcarriers so that the sequence P(k)is halved with respect to a DC subcarrier, and transmitting the asynchronization signal in the subcarriers.

In another aspect, a method of transmitting a synchronization signal ina wireless communication system includes generating a sequence P(k) fora synchronization signal from a Zadoff-Chu (ZC) sequence having thelength N=63 according to

${P(k)} = {\exp\left\{ {- \frac{j\;\pi\;{{Mk}\left( {k + 1} \right)}}{63}} \right\}}$

where M is a root index and k=0, 1, . . . , 30, 32, . . . , 62, wherebythe length of the sequence P(k) is N−1 which is an even number, mappingthe sequence P(k) to subcarriers s(n) so that the sequence P(k) ishalved with respect to a DC subcarrier at which the index n is zero asshowns(n)=P(k)

where n=k−31 and k=0, 1, . . . , 30, 32, . . . , 62, and transmittingthe synchronization signal on the subcarriers.

In still another aspect, a method of acquiring synchronization with acell in a wireless communication system includes receiving a primarysynchronization signal, and receiving a secondary synchronizationsignal. A sequence P(k) for the primary synchronization signal can begenerated from a ZC sequence having the odd numbered length N. Thesequence P(k) can have the even numbered length N−1 by omitting thecenter element of the ZC sequence and the sequence P(k) can be mapped tosubcarriers so that the sequence P(k) is halved with respect to a DCsubcarrier.

Advantageous Effects

Time/frequency ambiguity caused by a synchronization error can beavoided, and sequence detection errors can be decreased. In addition, asequence having both a good peak-to-average power ratio (PAPR) and agood correlation can be obtained.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph showing an example of time/frequency ambiguity of aZadoff-Chu (ZC) sequence.

FIG. 2 shows a structure of a synchronization channel.

FIG. 3 is a flowchart showing a sequence allocation method according toan embodiment of the present invention.

FIG. 4 is a flowchart showing a sequence allocation method according toan embodiment of the present invention.

FIG. 5 is a flowchart showing a sequence allocation method according toan embodiment of the present invention.

FIG. 6 is a flowchart showing a sequence allocation method according toan embodiment of the present invention.

FIG. 7 is a graph for comparing sensitivity of time/frequency ambiguity.

FIG. 8 is a graph showing an auto-correlation value of a sequence withrespect to three indices selected according to the conventional method.

FIG. 9 is a graph showing an auto-correlation value of a sequence withrespect to three indices selected according to the proposed method.

FIG. 10 is a graph showing a cross-correlation value of a sequence withrespect to three indices selected according to the conventional method.

FIG. 11 is a graph showing a cross-correlation value of a sequence withrespect to three indices selected according to the proposed method.

FIG. 12 shows an example of mapping a sequence having a length of N=63.

FIG. 13 shows another example of sequence mapping in comparison withFIG. 12.

FIG. 14 shows an example of mapping a sequence having a length of N=63.

FIG. 15 shows another example of sequence mapping in comparison withFIG. 14.

FIG. 16 shows an example of mapping a sequence having a length of N=63.

FIG. 17 shows another example of sequence mapping in comparison withFIG. 16.

FIG. 18 shows an example of mapping a sequence having a length of N=63.

FIG. 19 shows another example of sequence mapping in comparison withFIG. 18.

FIG. 20 shows an example of mapping a sequence having a length of N=65.

FIG. 21 shows another example of sequence mapping in comparison withFIG. 20.

FIG. 22 shows an example of mapping a sequence having a length of N=65.

FIG. 23 shows another example of sequence mapping in comparison withFIG. 22.

MODE FOR THE INVENTION

A wireless communication system includes a user equipment (UE) and abase station (BS). The UE may be fixed or mobile, and may be referred toas another terminology, such as a mobile station (MS), a user terminal(UT), a subscriber station (SS), a wireless device, etc. The BS isgenerally a fixed station that communicates with the UE and may bereferred to as another terminology, such as a node-B, a base transceiversystem (BTS), an access point, etc. There are one or more cells withinthe coverage of the BS. Hereinafter, downlink is defined as acommunication link from the BS to the UE, and uplink is defined as acommunication link from the UE to BS.

There is no limit in a multiple access scheme used in the wirelesscommunication system. The multiple access scheme may be based on avariety of multiple access schemes such as code division multiple access(CDMA), time division multiple access (TDMA), frequency divisionmultiple access (FDMA), single-carrier FDMA (SC-FDMA), orthogonalfrequency division multiple access (OFDMA), etc. For clear explanations,the following description focuses on an OFDMA-based wirelesscommunication system.

In the wireless communication system, a sequence is widely used forvarious purposes such as signal detection, channel estimation,multiplexing, etc. An orthogonal sequence having a good correlation isused so that sequence detection can be easily achieved in a receiver.The orthogonal sequence may be a constant amplitude zeroauto-correlation (CAZAC) sequence.

A k-th element of a Zadoff-Chu (ZC) sequence belonging to the CAZACsequence can be expressed as below:

$\begin{matrix}{{MathFigure}\mspace{14mu} 1} & \; \\{{{P(k)} = {\exp\left\{ {- \frac{{j\pi}\;{{Mk}\left( {k + 1} \right)}}{N}} \right\}}},{{for}\mspace{14mu} N\mspace{14mu}{odd}}} & \left\lbrack {{Math}.\mspace{14mu} 1} \right\rbrack \\{{{P(k)} = {\exp\left\{ {- \frac{j\;\pi\;{Mk}^{2}}{N}} \right\}}},{{for}\mspace{14mu} N\mspace{14mu}{even}}} & \;\end{matrix}$

where N denotes a length of a root ZC sequence, and M denotes a rootindex which is a relatively prime to N. If N is prime, the number ofroot indices of the ZC sequence is N−1.

A ZC sequence P(k) has three characteristics as follows.

$\begin{matrix}{{MathFigure}\mspace{14mu} 2} & \; \\{{{{P(k)}} = 1}{{{for}\mspace{14mu}{all}\mspace{14mu} k},N,M}} & \left\lbrack {{Math}.\mspace{14mu} 2} \right\rbrack \\{{MathFigure}\mspace{14mu} 3} & \; \\{{R_{M;N}(d)} = \left\{ \begin{matrix}{1,} & {{{for}\mspace{14mu} d} = 0} \\{0,} & {{{for}\mspace{14mu} d} \neq 0}\end{matrix} \right.} & \left\lbrack {{Math}.\mspace{14mu} 3} \right\rbrack \\{{MathFigure}\mspace{14mu} 4} & \; \\{{R_{{M_{1}M_{2}};N}(d)} = {{const}\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} M_{1,}M_{2}}} & \left\lbrack {{Math}.\mspace{14mu} 4} \right\rbrack\end{matrix}$

Equation 2 means that the ZC sequence always has a magnitude of one.Equation 3 means that auto-correlation of the ZC sequence is indicatedby a Dirac-delta function. The auto-correlation is based on circularcorrelation. Equation 4 means that cross correlation is always constant.

FIG. 1 is a graph showing an example of time/frequency ambiguity of a ZCsequence. The time/frequency ambiguity means that, when an offset occursin any one of time domain or frequency domain, the same offset occurs inanother domain by an amount corresponding to an index of the sequence.

Referring to FIG. 1, time/frequency ambiguity is shown when a frequencyoffset of 5 ppm is produced by generating and transmitting a ZC sequencehaving a length of N=64 and having an index M=1. Herein, the frequencyoffset of 5 ppm corresponds to a frequency offset of 10 kHz when acarrier frequency of 2 GHz is used, which represents a 2-part partialaperiodic auto-correlation in the time domain. It is assumed herein thata channel noise does not exist.

When the frequency offset is produced, an ambiguity peak is higher thana desired peak, and thus correct timing cannot be acquired. If thesequence is correlated in the frequency domain, locations may not becorrectly detected due to ambiguity when the time offset is produced.

If N=64, the number of available root sequences is 32. However, it isdifficult to use all 32 sequences due to the time/frequency ambiguity.

FIG. 2 shows a structure of a synchronization channel.

Referring to FIG. 2, a radio frame includes 10 subframes. One subframeincludes two slots. One slot includes a plurality of orthogonalfrequency division multiplexing (OFDM) symbols in the time domain.Although one slot includes 7 OFDM symbols in FIG. 2, the number of OFDMsymbols included in one slot may vary depending on a cyclic prefix (CP)structure. The structure of the radio frame is for exemplary purposesonly. Thus, the number of subframes included in the radio frame and thenumber of slots included in each subframe may change variously.

Primary synchronization channels (P-SCHs) are located in last OFDMsymbols of a 0-th slot and a 10-th slot. The same primarysynchronization signal (PSS) is used by two P-SCHs. The P-SCH is used toacquire OFDM symbol synchronization or slot synchronization. The PSS mayuse a Zadoff-Chu (ZC) sequence. Each of PSSs can represent a cellidentity according to a root index of the ZC sequence. When three PSSsexist, a BS selects one of the three PSSs, and transmits the selectedPSS by carrying the PSS on the last OFDM symbols of the 0-th slot andthe 10-th slot.

Secondary synchronization channels (S-SCHs) are located in OFDM symbolspositioned immediately before the last OFDM symbols of the 0-th slot andthe 10-th slot. The S-SCH and the P-SCH may be located in contiguousOFDM symbols. The S-SCH is used to acquire frame synchronization. OneS-SCH uses two secondary synchronization signals (SSSs). One S-SCHincludes two pseudo-noise (PN) sequences (i.e., m-sequences). Forexample, if one S-SCH includes 64 sub-carriers, two PN sequences havinga length of 31 are mapped to one S-SCH.

Locations or the number of OFDM symbols in which the P-SCH and the S-SCHare arranged over a slot are shown in FIG. 2 for exemplary purposesonly, and thus may vary depending on systems.

FIG. 3 is a flowchart showing a sequence allocation method according toan embodiment of the present invention.

Referring to FIG. 3, a length L of a mapping section for mapping asequence is determined (step S110). The mapping section may be inassociation with a data channel for transmitting user data or a controlchannel for transmitting a control signal. The mapping section may be inassociation with a radio resource for carrying data. The mapping sectionmay be a specific section including a plurality of subcarriers.

A length N of the sequence is determined (step S120). The sequencelength N may be less (or greater) than the length L of the mappingsection. According to one embodiment, when the length L of the mappingsection is even, a first odd number greater than the length L of themapping section may be selected as the sequence length N. Alternatively,a first odd number less than the length L of the mapping section may beselected as the sequence length N. The reason of selecting the oddnumber is that, when an even length is used to generate the ZC sequence,a correlation and an intrinsic characteristic of the sequence aresuperior to those obtained when an odd length is used. According toanother embodiment, a first even number greater than the length L of themapping section may be selected as the sequence length N. Alternatively,a first even number less than the length L of the mapping section may beselected as the sequence length N. If the length L of the mappingsection is odd, an even number may be selected as the sequence length N.According to still another embodiment, the sequence length N may begreater than the length L of the mapping section by 1. Alternatively,the sequence length N may be less than the length L of the mappingsection by 1. As such, a sequence characteristic (or correlationcharacteristic) can be improved when the sequence is assigned to themapping section while the sequence is adjusted so that the sequencelength N is different from the length L of the mapping section by 1.

The sequence is adjusted to fit the length L of the mapping section(step S130). If the sequence length N is less than the length L of themapping section, a null value (e.g., zero), an arbitrary value, a cyclicprefix, or a cyclic suffix may be inserted to a duration whose lengthexceeds the sequence length L. If the sequence length N is greater thanthe length L, an arbitrary element included in the sequence can beremoved. For example, removal may be performed starting from a lastportion of the sequence.

The sequence is mapped to the mapping section (step S140). If a directcurrent (DC) component exists in the mapping section, the DC componentcan be punctured. That is, the sequence is repeatedly mapped to themapping section, and elements corresponding to the DC component arereplaced with null values. Alternatively, the sequence may be mapped tothe mapping section except for the DC component. The DC componentrepresents a center frequency or a point where a frequency offset iszero in a baseband.

Although it has been exemplified that the sequence is mapped to themapping section after adjusting the sequence length to fit the length ofthe mapping section, the present invention is not limited thereto. Thus,the sequence length may be adjusted to fit the length of the mappingsection after the sequence is mapped to the mapping section.

In the OFDM/OFMDA system, the sequence is mapped to subcarriers in thefrequency domain. When a single carrier is used in transmission, forexample, when an SC-FDMA system is used, the sequence is mapped totime-domain samples. Sequences used as a pilot or ZC sequence-basedcontrol channels may be directly mapped in the frequency domain.

FIG. 4 is a flowchart showing a sequence allocation method according toan embodiment of the present invention.

Referring to FIG. 4, a length of a synchronization channel is determinedto L=64 when a DC subcarrier is included (step S210). Thesynchronization channel may be a P-SCH.

A length N of a sequence to be mapped is determined to a first oddnumber greater than the length L (step S220). Since L=64, it isdetermined to N=65.

The sequence is adjusted according to the length L of thesynchronization channel (step S230). In order to adjust the sequence tofit the length L, an arbitrary element included in the sequence isremoved. Herein, a last element of the sequence is removed.

The sequence is mapped to the synchronization channel (step S240). TheDC subcarrier is also included in the sequence mapping. Although thereis no restriction on a mapping order, continuous mapping is preferred tomaintain a CAZAC characteristic. The produced sequence may be cyclicshifted in the mapping.

An element of the sequence mapped to the DC subcarrier is punctured(step S250).

Since the sequence length is determined to be greater than the length ofthe synchronization channel while the DC subcarrier is punctured, theCAZAC characteristic can be maintained to the maximum extent possible inthe time domain of the ZC sequence inserted in the frequency domain. TheZC sequence has a duality relation in the time/frequency domains. Inaddition, effective correlation can be implemented by selecting rootindices (e.g., M=1 and M=63) of a symmetric pair. The symmetric pairrepresents a sequence having a pair of root indices whose sum is equalto the length of the sequence.

The produced sequence is effective for the synchronization channel usedfor synchronization between the UE and the BS or used for cell search.When the sequence length is selected to be an odd number instead of aprime number, the total number of indices of the sequence may bereduced. However, in this case, the sequence length can be determined ina more flexible manner.

FIG. 5 is a flowchart showing a sequence allocation method according toan embodiment of the present invention.

Referring to FIG. 5, a length of a synchronization channel is determinedto L=64 when a DC subcarrier is included (step S310). Thesynchronization channel may be a P-SCH.

A length N of a sequence to be mapped is determined to a first oddnumber less than the length L (step S320). Since L=64, it is determinedto N=63.

The sequence is adjusted according to the length L of thesynchronization channel (step S330). In order to adjust the sequence tofit the length L, an arbitrary element is added to the sequence. Theadded element may be a null value, an arbitrary value, a cyclic prefix,or a cyclic suffix. The added element may be inserted after the sequenceis cyclic shifted.

The sequence is mapped to the synchronization channel (step S340). TheDC subcarrier is also included in the sequence mapping. Although thereis no restriction on a mapping order, continuous mapping is preferred tomaintain a CAZAC characteristic. The produced sequence may be cyclicshifted in the mapping.

An element of the sequence mapped to the DC subcarrier is punctured(step S350).

Although it has been exemplified that the sequence is mapped to thesynchronization channel after adjusting the sequence length to fit thelength of the synchronization channel, the present invention is notlimited thereto. Thus, the sequence length may be adjusted to fit thelength of the synchronization channel after the sequence is mapped tothe synchronization channel.

FIG. 6 is a flowchart showing a sequence allocation method according toan embodiment of the present invention.

Referring to FIG. 6, a length of a synchronization channel is determinedto L=64 when DC subcarrier is included (step S410). The synchronizationchannel may be a P-SCH.

A length N of a sequence to be mapped is determined to a first oddnumber less than the length L (step S420). Since L=64, it is determinedto N=63.

The sequence is mapped to the synchronization channel except for the DCsubcarrier (step S430). A null value is assigned to the DC subcarrier.In this case, a CAZAC characteristic may not be maintained due toambiguity in the DC subcarrier.

Now, a simulation result obtained by comparing the proposed method andthe conventional method will be described. It will be assumed that asynchronization channel has a length of L=64. In the conventionalmethod, a ZC sequence having a length of N=64 is mapped to thesynchronization channel without performing a specific process. In theproposed method, a ZC sequence having a length of N=65 is mapped to thesynchronization channel and a DC subcarrier is punctured.

FIG. 7 is a graph for comparing sensitivity of time/frequency ambiguity.The graph shows the sensitivity of time/frequency ambiguity in a casewhere a frequency offset is 5 ppm and a 2-part partial aperiodicauto-correlation is used. The sensitivity of time/frequency ambiguity isa ratio between an ambiguity peak and a desired peak. The less thesensitivity, the better the characteristic.

Referring to FIG. 7, when the proposed method is used, the sequence hasa lower sensitivity than when the conventional method is used. A lowestsensitivity is about 0.65 according to the conventional method and about0.3 according to the proposed method.

Assume that three sequences are used for a PSS. In FIG. 7, according tothe conventional method, the lowest sensitivity is obtained when M=31,33, and 29. Thus, these values are selected. According to the proposedmethod, the lowest sensitivity is obtained when M=34, 31, and 38. Thus,these values are selected.

FIG. 8 is a graph showing an auto-correlation value of a sequence withrespect to three indices selected according to the conventional method.FIG. 9 is a graph showing an auto-correlation value of a sequence withrespect to three indices selected according to the proposed method.

Referring to FIGS. 8 and 9, second or later peaks appearing in thesequence selected according to the proposed method is relatively lowerthan those appearing in the sequence selected according to theconventional method, and show a significant difference from a firstpeak.

According to the proposed method, a possibility of finding correcttiming is further increased even in a condition where a frequency offsetor a time offset exists. Therefore, a better characteristic can be shownby using a sequence used in a synchronization channel forsynchronization.

FIG. 10 is a graph showing a cross-correlation value of a sequence withrespect to three indices selected according to the conventional method.FIG. 11 is a graph showing a cross-correlation value of a sequence withrespect to three indices selected according to the proposed method. Thecross-correlation value represents an interference level betweensequences having different indices. The less the average and thedispersion, the better the performance.

Referring to FIGS. 10 and 11, according to the conventional method, theaverage of the cross-correlation values is about 0.522 and thedispersion thereof is 0.200. Whereas, according to the proposed method,the average is 0.503 and the dispersion is 0.195.

The proposed method can be applied not only to the synchronizationchannel but also to various other types of radio resources.

According to one embodiment, a resource block is taken into account. Theresource block includes a plurality of subcarriers. For example, oneresource block may include 12 subcarriers, and 10 resource blocks may beallocated to a mapping section. In this case, a length of the mappingsection is L=120.

First, assume that a first odd number, i.e., N=121, greater than thelength, i.e., L=120, of a desired mapping section is selected as asequence length N. One arbitrary element of the sequence having a lengthof 121 is removed, and then the sequence is mapped to 120 subcarriers.In this case, mapping is performed irrespective of an insertion order.However, continuous mapping is preferred to maintain a CAZACcharacteristic. The produced sequence may be cyclic shifted in themapping.

Next, assume that a first odd number, i.e., N=119, less than a length,i.e., L=120, of the desired mapping section is selected as the sequencelength N. A sequence having a length of 119 is produced, and oneremaining subcarrier among 120 subcarriers originally intended to beused may be inserted with a null value or an arbitrary value or may beappended with a cyclic prefix or a cyclic suffix. The cyclic prefix orthe cyclic suffix may be appended after cyclic shifting is performed. Inaddition, discontinuous mapping is also possible by inserting the nullvalue in the middle of the sequence.

According to another embodiment, assume that one resource block includes12 subcarriers, and two resource blocks are allocated to a mappingsection. In this case, a length of the mapping section is L=24.

First, assume that a first odd number, i.e., N=25, greater than thelength, i.e., L=24, of a desired mapping section is selected as asequence length N. One arbitrary element of the sequence having a lengthof 25 is removed, and then the sequence is mapped to 24 subcarriers. Inthis case, mapping is performed irrespective of an insertion order.However, continuous mapping is preferred to maintain a CAZACcharacteristic. The produced sequence may be cyclic shifted in themapping.

Next, assume that a first odd number, i.e., N=23, less than a length,i.e., L=24, of the desired mapping section is selected as the sequencelength N. A sequence having a length of 23 is produced, and oneremaining subcarrier among 24 subcarriers originally intended to be usedmay be inserted with a null value or an arbitrary value or may beappended with a cyclic prefix or a cyclic suffix. The cyclic prefix orthe cyclic suffix may be appended after cyclic shifting is performed. Inaddition, discontinuous mapping is also possible by inserting the nullvalue in the middle of the sequence.

According to still another embodiment, a mapping section may have anarbitrary length. For example, the mapping section may include 780subcarriers (i.e., L=780). First, assume that a first odd number, i.e.,N=781, greater than the length, i.e., L=780, of a desired mappingsection is selected as a sequence length N. One arbitrary element of thesequence having a length of 781 is removed, and then the sequence ismapped to 780 subcarriers. In this case, mapping is performedirrespective of an insertion order. However, continuous mapping ispreferred to maintain a CAZAC characteristic. The produced sequence maybe cyclic shifted in the mapping.

Next, assume that a first odd number, i.e., N=779, less than a length,i.e., L=780, of the desired mapping section is selected as the sequencelength N. A sequence having a length of 779 is produced, and oneremaining subcarrier among 780 subcarriers originally intended to beused may be inserted with a null value or an arbitrary value or may beappended with a cyclic prefix or a cyclic suffix. The cyclic prefix orthe cyclic suffix may be appended after cyclic shifting is performed. Inaddition, discontinuous mapping is also possible by inserting the nullvalue in the middle of the sequence.

<Sequence Mapping>

Hereinafter, a sequence mapping method will be described according to anembodiment of the present invention. For clear explanations, it will beassumed that a length of a mapping section is L=64. A ZC sequence ismapped to 64 subcarriers including a DC subcarrier.

FIG. 12 shows an example of mapping a sequence having a length of N=63.Herein, a fast Fourier transform (FFT) window has a size Nf=64.

Referring to FIG. 12, a 0-th element P(0) of a ZC sequence is mapped toa DC subcarrier, and then all elements located right to the DCsubcarrier are sequentially mapped to the remaining subcarriers. Asubcarrier adjacent left to the DC subcarrier is mapped to a 62-thelement P(62). A null value is inserted to a subcarrier (herein, a 32-thsubcarrier) to which the sequence is not mapped to the mapping section.The 0-th element P(0) mapped to the DC subcarrier is punctured asindicated by a dotted line.

Herein, for convenience, one side of the DC subcarrier is defined as theleft side, and the opposite side thereof is defined as the right side.However, the left side and the right side may be differently defined andthus are not limited as shown in the figure.

FIG. 13 shows another example of sequence mapping in comparison withFIG. 12.

Referring to FIG. 13, in comparison with the example of FIG. 12, an FFTwindow has a size Nf=128. With a DC subcarrier being located in thecenter, the sequence is mapped to subcarriers in the same manner as whenthe FFT window size is 64, and null values are inserted to the remainingsubcarriers.

FIG. 14 shows an example of mapping a sequence having a length of N=63.

Referring to FIG. 14, a mapping section has a length L=64, and asequence has a length N=63. A sequence P(k) is produced from afrequency-domain ZC sequence having a sequence length N=63, which isexpressed as below:

$\begin{matrix}{{MathFigure}\mspace{14mu} 5} & \; \\{{P(k)} = {\exp\left\{ {- \frac{j\;\pi\;{{Mk}\left( {k + 1} \right)}}{63}} \right\}}} & \left\lbrack {{Math}.\mspace{14mu} 5} \right\rbrack\end{matrix}$

where M denotes a root index, and k=0, 1, . . . , 30, 32, . . . , 62.The reason of excluding a center element (i.e., k=31) is to remove a DCsubcarrier from a mapping section s(n) and also to halve the sequence tobe mapped as expressed as below:MathFigure 6s(n)=P(k)  [Math.6]

where n=k−31.

That is, with the DC subcarrier being located in the center, one half ofthe sequence is mapped to left 31 subcarriers and the other half ismapped to right 31 subcarriers. P(0) is mapped to a leftmost subcarriera(−31). Then, the sequence is sequentially mapped except for the DCsubcarrier.

A center element is omitted from a ZC sequence having an odd length toproduce a sequence having an even length. The produced sequence ishalved with respect to the DC subcarrier and is then mapped.Accordingly, a characteristic of the ZC sequence can be maintained inthe time domain even if the ZC sequence is mapped in the frequencydomain, which will be described later. A root-symmetry property and acentral-symmetry property are satisfied in the time domain.

FIG. 15 shows another example of sequence mapping in comparison withFIG. 14.

Referring to FIG. 15, in comparison with the example of FIG. 14, an FFTwindow has a size Nf=128. With a DC subcarrier being located in thecenter, the sequence is mapped to subcarriers in the same manner as whenthe FFT window size is 64, and null values are inserted to the remainingsubcarriers.

FIG. 16 shows an example of mapping a sequence having a length of N=63.Herein, an FFT window has a size Nf=64.

Referring to FIG. 16, a 0-th element P(0) of a ZC sequence is mapped toa DC subcarrier, and then all elements located right to the DCsubcarrier are sequentially mapped to the remaining subcarriers. Asubcarrier adjacent left to the DC subcarrier is mapped to a 62-thelement P(62). Unlike the example of FIG. 12, a null value is notinserted to a subcarrier (herein, a 32-th subcarrier) to which thesequence is not mapped to the mapping section. Instead, a 31-th elementP(31) is copied and inserted. That is, the sequence can be extendedthrough cyclic extension when the mapping section is not sufficient. The0-th element P(0) mapped to the DC subcarrier is punctured as indicatedby a dotted line.

FIG. 17 shows another example of sequence mapping in comparison withFIG. 16

Referring to FIG. 17, in comparison with the example of FIG. 16, an FFTwindow has a size Nf=128. With a DC subcarrier being located in thecenter, the sequence is mapped to subcarriers in the same manner as whenthe FFT window size is 64, and null values are inserted to the remainingsubcarriers.

FIG. 18 shows an example of mapping a sequence having a length of N=63.Herein, an FFT window has a size Nf=64.

Referring to FIG. 18, a ZC sequence is sequentially mapped, startingfrom a leftmost subcarrier, so that a center element (herein, a 31-thelement P(31)) of the ZC sequence is mapped to a DC subcarrier. Thesequence P(31) mapped to the DC subcarrier is punctured.

A last element P(62) is copied and inserted to a subcarrier (herein, a32-th subcarrier) to which the sequence cannot be mapped in a mappingsection. That is, the sequence can be extended by cyclic extension ifthe mapping section is not sufficient.

Accordingly, a characteristic of the ZC sequence can be maintained inthe time domain even if the ZC sequence is mapped in the frequencydomain, which will be described later. The root-symmetry property andthe central-symmetry property are satisfied in the time domain.

FIG. 19 shows another example of sequence mapping in comparison withFIG. 18

Referring to FIG. 19, in comparison with the example of FIG. 18, an FFTwindow has a size Nf=128. With a DC subcarrier being located in thecenter, the sequence is mapped to subcarriers in the same manner as whenthe FFT window size is 64, and null values are inserted to the remainingsubcarriers.

FIG. 20 shows an example of mapping a sequence having a length of N=65.Herein, an FFT window has a size Nf=64.

Referring to FIG. 20, a 0-th element P(0) of a ZC sequence is mapped toa DC subcarrier, and then all elements located right to the DCsubcarrier are sequentially mapped to the remaining subcarriers. Asubcarrier adjacent left to the DC subcarrier is mapped to a 63-thelement P(63). A 64th element P(64) of the ZC sequence is a remainingelement, and is thus truncated. The 0-th element P(0) mapped to the DCsubcarrier is punctured as indicated by a dotted line.

FIG. 21 shows another example of sequence mapping in comparison withFIG. 20.

Referring to FIG. 21, in comparison with the example of FIG. 20, an FFTwindow has a size Nf=128. With a DC subcarrier being located in thecenter, the sequence is mapped to subcarriers in the same manner as whenthe FFT window size is 64, and null values are inserted to the remainingsubcarriers.

FIG. 22 shows an example of mapping a sequence having a length of N=65.Herein, an FFT window has a size Nf=64.

Referring to FIG. 22, a ZC sequence is sequentially mapped, startingfrom a leftmost subcarrier, so that a center element (herein, a 32-thelement P(32)) of the ZC sequence is mapped to a DC subcarrier. A 64-thelement P(64) of the ZC sequence is a remaining element, and is thustruncated. The sequence P(32) mapped to the DC subcarrier is punctured.

Accordingly, a characteristic of the ZC sequence can be maintained inthe time domain even if the ZC sequence is mapped in the frequencydomain, which will be described later. The root-symmetry property andthe central-symmetry property are satisfied in the time domain.

FIG. 23 shows another example of sequence mapping in comparison withFIG. 22.

Referring to FIG. 23, in comparison with the example of FIG. 22, an FFTwindow has a size Nf=128. With a DC subcarrier being located in thecenter, the sequence is mapped to subcarriers in the same manner as whenthe FFT window size is 64, and null values are inserted to the remainingsubcarriers.

<Verification on Whether Characteristic of ZC Sequence is Maintained>

As described above with reference to FIG. 14, 18, or 22, when a sequenceis mapped to subcarriers such that a center element of a ZC sequencecorresponds to a DC subcarrier, the root-symmetry property and thecentral-symmetry property are satisfied in the time domain.

The root-symmetry property means that to or more root sequences have aspecific relation and thus show a specific relation with a certain rootsequence index. Requirements for satisfying the root-symmetry propertyare shown:MathFigure 7m1+m2=(½·N)·n or m1−m2=±(½·N)·n  [Math.7]

where n=1, 2, . . . , m1 and m2 denote root sequence indices, and Ndenotes a sequence length.

For example, if N is odd, it means that a conjugate symmetry propertyappears.

When considering a root sequence p^(m1) having an index m1 and anotherroot sequence p^(m2) having an index m2=N−m1, the conjugate symmetryproperty in both the time/frequency domains can be expressed as shownMathFigure 8p ^(m1)(n)=(p ^(m2)(n))*  [Math.8]

where ( )* denotes a conjugate. For example, m1=29 and m2=34=N−m1=63−29have the root-symmetry relation with each other.

If N is even, one root index has a special conjugate relation with theother root index For example, if indices 1, 17, 19, and 35 are selectedwhen N=36, the root-symmetry relation can be expressed as shown below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 9} & \; \\{{{p^{{m\; 0} = 1}(k)} = {\exp\left( {{- {j\pi}} \cdot 1 \cdot \frac{k^{2}}{36}} \right)}}\begin{matrix}{{p^{{m\; 1} = 17}(k)} = {\exp\left( {{- {j\pi}}\; 17\;\frac{k^{2}}{36}} \right)}} \\{= {\exp\left( {{- j}\;{\pi\left( {18 - 1} \right)}\frac{k^{2}}{36}} \right)}} \\{= {\exp\left( {- {j\left( {{\frac{\pi}{2}k^{2}} - {\frac{\pi}{36}k^{2}}} \right)}} \right)}} \\{= \left\{ \begin{matrix}{\left( {a_{even}^{{m\; 0} = 1}(k)} \right)^{*},} & {{when}\mspace{14mu} k\mspace{14mu}{is}\mspace{14mu}{even}} \\{{{- j} \cdot \left( {a_{odd}^{{m\; 0} = 1}(k)} \right)^{*}},} & {otherwise}\end{matrix} \right.}\end{matrix}\begin{matrix}{{p^{{m\; 2} = 19}(k)} = {\exp\left( {{- {j\pi}}\; 19\frac{k^{2}}{36}} \right)}} \\{= {\exp\left( {{- {{j\pi}\left( {18 + 1} \right)}}\frac{k^{2}}{36}} \right)}} \\{= {\exp\left( {- {j\left( {{\frac{\pi}{2}k^{2}} + {\frac{\pi}{36}k^{2}}} \right)}} \right)}} \\{= \left\{ \begin{matrix}{\left( {a_{even}^{{m\; 0} = 1}(k)} \right),} & {{when}\mspace{14mu} k\mspace{14mu}{is}\mspace{14mu}{even}} \\{{{- j} \cdot \left( {a_{odd}^{{m\; 0} = 1}(k)} \right)},} & {otherwise}\end{matrix} \right.}\end{matrix}\begin{matrix}{{p^{{m\; 3} = 35}(k)} = {\exp\left( {{- {j\pi}}\; 35\frac{k^{2}}{36}} \right)}} \\{= {\exp\left( {{- {j\pi}}\;\left( {36 - 1} \right)\frac{k^{2}}{36}} \right)}} \\{= {\exp\left( {- {j\left( {{\pi\; k^{2}} - {\frac{\pi}{36}k^{2}}} \right)}} \right)}} \\{= \left\{ \begin{matrix}{\left( {a_{even}^{{m\; 0} = 1}(k)} \right)^{*},} & {{when}\mspace{14mu} k\mspace{14mu}{is}\mspace{14mu}{even}} \\{{{- j} \cdot \left( {a_{odd}^{{m\; 0} = 1}(k)} \right)^{*}},} & {otherwise}\end{matrix} \right.}\end{matrix}} & \left\lbrack {{Math}.\mspace{14mu} 9} \right\rbrack\end{matrix}$

The central-symmetry property means that a signal in the time domain hasa characteristic as expressed below:MathFigure 10p ^(m)(n)=p ^(m)(N _(f) −n)  [Math.10]

where p^(m)(n) denotes an n-th element of a ZC sequence having an indexm, and Nf denotes an FFT window size. When the central-symmetry propertyof Equation 10 is satisfied, the root-symmetry property also can besatisfied.

(1) Requirements for Satisfying Root-Symmetry Property

It will be assumed herein that a ZC sequence has an odd length whenproduced. In addition, a length of the ZC sequence is N and an FFTwindow size is Nf, where N<=Nf. First, when the ZC sequence is insertedin the frequency domain, the ZC sequence is related to a signalconverted to the time domain, as described below.

A ZC sequence P^(m)(k) having an index m in the frequency domain can beconverted to the time domain according to the FFT window size Nf asexpressed as below:

$\begin{matrix}{{MathFigure}\mspace{14mu} 11} & \; \\\begin{matrix}{{p^{m}(n)} = {\frac{1}{N_{f}}{\sum\limits_{k = 0}^{N_{f} - 1}{{P^{m}(k)}W^{- {kn}}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = 0}^{N_{f} - 1}{{\exp\left( {{- {j\pi}}\;{{{mk}\left( {k + 1} \right)}/N}} \right)}W^{- {kn}}}}}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 11} \right\rbrack\end{matrix}$

where n=0, 1, 2, . . . , Nf−1, and W=exp(−2jπ/Nf). After converting theZC sequence having an index M=N−m in the frequency domain to the timedomain according to the FFT window size Nf, the ZC sequence isconjugated as expressed as below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 12} & \; \\\begin{matrix}{\left( {p^{N - m}(n)} \right)^{*} = \left( {\frac{1}{N_{f}}{\sum\limits_{k = 0}^{N_{f} - 1}{{P^{N - m}(k)}W^{- {kn}}}}} \right)^{*}} \\{= \left( {\frac{1}{N_{f}}{\sum\limits_{k = 0}^{N_{f} - 1}{{\exp\begin{pmatrix}{{- j}\;{\pi\left( {N - m} \right)}} \\{{k\left( {k + 1} \right)}/N}\end{pmatrix}}W^{- {kn}}}}} \right)^{*}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = 0}^{N_{f} - 1}{{\exp\left( {{- {j\pi}}\;{{{mk}\left( {k + 1} \right)}/N}} \right)}W^{- {kn}}}}}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 12} \right\rbrack\end{matrix}$

In addition, Equation 13 below is satisfied.

$\begin{matrix}{{MathFigure}\mspace{14mu} 13} & \; \\\begin{matrix}{\left( {p^{N - m}\left( {N_{f} - n} \right)} \right)^{*} = \left( {\frac{1}{N_{f}}{\sum\limits_{k = 0}^{N_{f} - 1}{{P^{N - m}(k)}W^{- {k{({N_{f} - n})}}}}}} \right)^{*}} \\{= \left( {\frac{1}{N_{f}}{\sum\limits_{k = 0}^{N_{f} - 1}{{\exp\begin{pmatrix}{- {{j\pi}\left( {N - m} \right)}} \\{{k\left( {k + 1} \right)}/N}\end{pmatrix}}W^{- {kn}}}}} \right)^{*}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = 0}^{N_{f} - 1}{{\exp\left( {{- {j\pi}}\;{{{mk}\left( {k + 1} \right)}/N}} \right)}W^{- {kn}}}}}} \\{= {p^{m}(n)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 13} \right\rbrack\end{matrix}$

According to Equation 12 and Equation 13 above, a necessary andsufficient condition for satisfying the root-symmetry property in thetime domain can be expressed as below.MathFigure 14p ^(N-m)(n)=p ^(N-m)(N _(f) −n) or p ^(m)(n)=p ^(m)(N _(f)−n)  [Math.14]

Equation 14 represents the central-symmetry property.

(2) Puncture Location for Satisfying Root-Symmetry Property andCentral-Symmetry Property

First, a specific duration to be punctured in the frequency domain whilemaintaining the root-symmetry property will be described. It will beassumed herein that a signal satisfies the central-symmetry property andmaintains the root-symmetry property before puncturing. In addition, itis also assumed that N<=Nf.

A value k′ satisfying the above mentioned requirement can be expressedby Equation 15 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 15} & \; \\{{{p^{m}(n)} - {\frac{1}{N_{f}}{\exp\left( {{- j}\;\pi\;{{{mk}^{\prime}\left( {k^{\prime} + 1} \right)}/N}} \right)}{\exp\left( {j\; 2\pi\; k^{\prime}{n/N_{f}}} \right)}}} = \left( {\left( {p^{N - m}(n)} \right)^{*} - \left( {\frac{1}{N_{f}}{\exp\begin{pmatrix}{{- j}\;{\pi\left( {N - m} \right)}} \\{{k^{\prime}\left( {k^{\prime} + 1} \right)}/N}\end{pmatrix}}{\exp\left( {j\; 2\pi\; k^{\prime}{n/N_{f}}} \right)}} \right)^{*}} \right.} & \left\lbrack {{Math}.\mspace{14mu} 15} \right\rbrack\end{matrix}$

A discarded single carrier component can be expressed in the time domainas a negative term. Equation 15 can be simplified by Equation 16 below.MathFigure 16exp(4πk′n/N _(f))=1  [Math.16]

Accordingly, the value k′ can be expressed by Equation 17 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 17} & \; \\{{k^{\prime} = 0},{\pm \frac{N_{f}}{2}},{\pm N_{f}},\ldots} & \left\lbrack {{Math}.\mspace{14mu} 17} \right\rbrack\end{matrix}$

When a periodical characteristic of FFT is considered, the desired valuek′ in a corresponding duration is k′=0, ±Nf/2. It can be seen that apuncture location depends on the FFT window size Nf if the sequencesatisfies the root-symmetry property before puncturing.

(3) Verification on Root-Symmetry and Central-Symmetry for the Exampleof FIG. 14 or FIG. 15

In the example of FIG. 14 or FIG. 15, the ZC sequence is used in thefrequency domain, and thus Equation 18 below is satisfied.MathFigure 18P ^(m)(k)=P ^(m)(N−k−1)  [Math.18]

When a mapping relation used in the figure is considered, an assignedsequence D^(m) (k) can be expressed by Equation 19 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 19} & \; \\{{D^{m}(k)} = \left\{ {{\begin{matrix}{0,} & {k = {- 32}} \\{{P^{m}\left( {k + 31} \right)},} & {{k = {- 31}},\ldots\mspace{14mu},{- 1}} \\{0,} & {k = 0} \\{{P^{m}\left( {k + 31} \right)},} & {{k = 1},\ldots\mspace{11mu},31}\end{matrix}{or}{D^{m}(k)}} = \left\{ \begin{matrix}{0,} & {k = {- 32}} \\{{P^{m}\left( {N - \left( {k + 31} \right) - 1} \right)},} & {{k = {- 31}},\ldots\mspace{14mu},{- 1}} \\{0,} & {k = 0} \\{{P^{m}\left( {N - \left( {k + 31} \right) - 1} \right)},} & {{k = 1},\ldots\mspace{11mu},31}\end{matrix} \right.} \right.} & \left\lbrack {{Math}.\mspace{14mu} 19} \right\rbrack\end{matrix}$

Therefore, Equation 20 below is satisfied.MathFigure 20D ^(m)(k)=D ^(m)(N _(f) −k)  [Math.20]

A time-domain signal d^(m)(n) can be expressed by Equation 21 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 21} & \; \\{{d^{m}(n)} = {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}\;{{D^{m}(k)}W^{- {kn}}}}}} & \left\lbrack {{Math}.\mspace{14mu} 21} \right\rbrack\end{matrix}$

In Equation 21, n=0, 1, 2, . . . , Nf−1, and W=exp(−2jπ/Nf).

A time-domain signal d^(m)(Nf−n) representing the central-symmetryproperty of Equation 10 above can be expressed by Equation 22 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 22} & \; \\\begin{matrix}{{d^{m}\left( {N_{f} - n} \right)} = {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{- {k{({N_{f} - n})}}}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{kn}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}\left( {N_{f} - k} \right)}W^{- {kn}}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{l = {N_{f} + 32}}^{N_{f} - 31}\;{{D^{m}(l)}W^{{({N_{f} - l})}n}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{l = {- 31}}^{32}{{D^{m}(l)}W^{{({N_{f} - l})}n}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{l = {- 31}}^{32}{{D^{m}(l)}W^{{- 1}\; n}}}}} \\{= {d^{m}(n)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 22} \right\rbrack\end{matrix}$

Equation 22 above shows that the central-symmetry property is satisfied.

In addition, Equation 23 below also shows that the root-symmetryproperty is satisfied.

$\begin{matrix}{{MathFigure}\mspace{14mu} 23} & \; \\\begin{matrix}{\left( {d^{m}\left( {N_{f} - n} \right)} \right)^{*} = \left( {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{N - m}(k)}W^{- {kn}}}}} \right)^{*}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{\left( {D^{N - m}(k)} \right)^{*}W^{kn}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{kn}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}\left( {N_{f} - k} \right)}W^{- {kn}}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{l = {N_{f} + 32}}^{N_{f} - 31}\;{{D^{m}(l)}W^{{({N_{f} - l})}n}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{l = {- 31}}^{32}{{D^{m}(l)}W^{- \ln}}}}} \\{= {d^{m}(n)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 23} \right\rbrack\end{matrix}$

(4) Verification on Root-Symmetry and Central-Symmetry for the Exampleof FIG. 18 or FIG. 19

In the example of FIG. 18 or FIG. 19, the ZC sequence is used in thefrequency domain, and thus Equation 18 above is satisfied.

When a mapping relation used in the figure is considered, an assignedsequence D^(m)(k) can be expressed by Equation 24 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 24} & \; \\{{D^{m}(k)} = \left\{ {{\begin{matrix}{{P^{m}(62)},} & {k = {- 32}} \\{{P^{m}\left( {k + 31} \right)},} & {{k = {- 31}},\ldots\mspace{14mu},{- 1}} \\{0,} & {k = 0} \\{{P^{m}\left( {k + 31} \right)},} & {{k = 1},\ldots\mspace{14mu},31}\end{matrix}{or}{D^{m}(k)}} = \left\{ \begin{matrix}{{P^{m}(62)},} & {k = {- 32}} \\{{P^{m}\left( {N - \left( {k + 31} \right) - 1} \right)},} & {{k = {- 31}},\ldots\mspace{14mu},{- 1}} \\{0,} & {k = 0} \\{{P^{m}\left( {N - \left( {k + 31} \right) - 1} \right)},} & {{k = 1},\ldots\mspace{14mu},31}\end{matrix} \right.} \right.} & \left\lbrack {{Math}.\mspace{14mu} 24} \right\rbrack\end{matrix}$

Therefore, Equation 25 below is satisfied.MathFigure 25D ^(m)(k)=D ^(m)(N _(f) −k), k≠−32  [Math.25]

A time-domain signal d^(m)(n) can be expressed by Equation 26 below:

$\begin{matrix}{{MathFigure}\mspace{14mu} 26} & \; \\\begin{matrix}{{d^{m}(n)} = {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{- {kn}}}}}} \\{= {\frac{1}{N_{f}}\left( {{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} + {\sum\limits_{k = {- 31}}^{31}{{D^{m}(k)}W^{- {kn}}}}} \right)}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{k = {- 31}}^{31}{{D^{m}(k)}W^{- {kn}}}}} \right)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 26} \right\rbrack\end{matrix}$

where n=0, 1, 2, . . . , Nf−1, and W=exp(−2jπ/Nf).

A time-domain signal d^(m)(Nf−n) representing the central-symmetryproperty of Equation 10 above can be expressed by Equation 27 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 27} & \; \\\begin{matrix}{{d^{m}\left( {N_{f} - n} \right)} = {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{- {k{({N_{f} - n})}}}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{kn}}}}} \\{= {\frac{1}{N_{f}}\begin{pmatrix}{{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} +} \\{\sum\limits_{k = {- 31}}^{31}{{D^{m}\left( {N_{f} - k} \right)}W^{- {kn}}}}\end{pmatrix}}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{l = {N_{f} + 31}}^{N_{f} - 31}\;{{D^{m}(l)}W^{{({N_{f} - l})}n}}}} \right)}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{l = {- 31}}^{31}\;{{D^{m}(l)}W^{{({N_{f} - l})}n}}}} \right)}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{l = {- 31}}^{31}{{D^{m}(l)}W^{{- l}\; n}}}} \right)}} \\{= {d^{m}(n)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 27} \right\rbrack\end{matrix}$

Equation 27 above shows that the central-symmetry property is satisfied.

Equation 28 below also shows that the root-symmetry property issatisfied.

$\begin{matrix}{{MathFigure}\mspace{14mu} 28} & \; \\\begin{matrix}{\left( {d^{N - m}(n)} \right)^{*} = \left( {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{N - m}(k)}W^{- {kn}}}}} \right)^{*}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{\left( {D^{N - m}(k)} \right)^{*}W^{kn}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{kn}}}}} \\{= {\frac{1}{N_{f}}\left( {{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} + {\sum\limits_{k = {- 31}}^{31}{{D^{m}(k)}W^{kn}}}} \right)}} \\{= {\frac{1}{N_{f}}\begin{pmatrix}{{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} +} \\{\sum\limits_{k = {- 31}}^{31}{{D^{m}\left( {N_{f} - k} \right)}W^{kn}}}\end{pmatrix}}} \\{= {\frac{1}{N_{f}}\begin{pmatrix}{{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} +} \\{\sum\limits_{l = {N_{f} + 31}}^{N_{f} - 31}{{D^{m}(l)}W^{{({N_{f} - l})}n}}}\end{pmatrix}}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{l = {- 31}}^{32}{{D^{m}(l)}W^{{- l}\; n}}}} \right)}} \\{= {d^{m}(n)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 28} \right\rbrack\end{matrix}$

(5) Verification on Root-Symmetry and Central-Symmetry for the Exampleof FIG. 22 or FIG. 23

In the example of FIG. 22 or FIG. 23, the ZC sequence is used in thefrequency domain, and thus Equation 18 above is satisfied.

When a mapping relation used in the figure is considered, an assignedsequence D^(m) (k) can be expressed by Equation 29 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 29} & \; \\{{D^{m}(k)} = \left\{ {{\begin{matrix}{{P^{m}\left( {k + 32} \right)},} & {{k = {- 32}},\ldots\mspace{14mu},{- 1}} \\{0,} & {k = 0} \\{{P^{m}\left( {k + 32} \right)},} & {{k = 1},\ldots\mspace{14mu},31}\end{matrix}{or}{D^{m}(k)}} = \left\{ \begin{matrix}{{P^{m}\left( {N - \left( {k + 32} \right) - 1} \right)},} & {{k = {- 32}},\ldots\mspace{14mu},{- 1}} \\{0,} & {k = 0} \\{{P^{m}\left( {N - \left( {k + 32} \right) - 1} \right)},} & {{k = 1},\ldots\mspace{14mu},31}\end{matrix} \right.} \right.} & \left\lbrack {{Math}.\mspace{14mu} 29} \right\rbrack\end{matrix}$

A last element P^(m)(64) is discarded. Therefore, Equation 30 below issatisfied.MathFigure 30D ^(m)(k)=D ^(m)(N _(f) −k), k≠−32  [Math.30]

A time-domain signal d^(m)(n) can be expressed by Equation 31 below:

$\begin{matrix}{{MathFigure}\mspace{14mu} 31} & \; \\\begin{matrix}{{d^{m}(n)} = {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{- {kn}}}}}} \\{= {\frac{1}{N_{f}}\left( {{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} + {\sum\limits_{k = {- 31}}^{31}{{D^{m}(k)}W^{- {kn}}}}} \right)}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{k = {- 31}}^{31}{{D^{m}(k)}W^{- {kn}}}}} \right)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 31} \right\rbrack\end{matrix}$

where n=0, 1, 2, . . . , Nf−1, and W=exp(−2jπ/Nf).

A time-domain signal d^(m)(Nf−n) representing the central-symmetryproperty of Equation 10 above can be expressed by Equation 32 below.

$\begin{matrix}{{MathFigure}\mspace{14mu} 32} & \; \\\begin{matrix}{{d^{m}\left( {N_{f} - n} \right)} = {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{- {k{({N_{f} - n})}}}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{kn}}}}} \\{= {\frac{1}{N_{f}}\begin{pmatrix}{{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} +} \\{\sum\limits_{k = {- 31}}^{31}{{D^{m}\left( {N_{f} - k} \right)}W^{kn}}}\end{pmatrix}}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{l = {N_{f} + 31}}^{N_{f} - 31}{{D^{m}(l)}W^{{({N_{f} - l})}n}}}} \right)}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{l = {- 31}}^{31}\;{{D^{m}(l)}W^{{({N_{f} - l})}n}}}} \right)}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{l = {- 31}}^{31}{{D^{m}(l)}W^{{- l}\; n}}}} \right)}} \\{= {d^{m}(n)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 32} \right\rbrack\end{matrix}$

Equation 32 above shows that the central-symmetry property is satisfied.

Equation 33 below also shows that the root-symmetry property issatisfied.

$\begin{matrix}{{MathFigure}\mspace{14mu} 33} & \; \\\begin{matrix}{\left( {d^{N - m}(n)} \right)^{*} = \left( {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{N - m}(k)}W^{- {kn}}}}} \right)^{*}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{\left( {D^{N - m}(k)} \right)^{*}W^{kn}}}}} \\{= {\frac{1}{N_{f}}{\sum\limits_{k = {- 32}}^{31}{{D^{m}(k)}W^{kn}}}}} \\{= {\frac{1}{N_{f}}\left( {{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} + {\sum\limits_{k = {- 31}}^{31}{{D^{m}(k)}W^{kn}}}} \right)}} \\{= {\frac{1}{N_{f}}\begin{pmatrix}{{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} +} \\{\sum\limits_{k = {- 31}}^{31}{{D^{m}\left( {N_{f} - k} \right)}W^{kn}}}\end{pmatrix}}} \\{= {\frac{1}{N_{f}}\begin{pmatrix}{{{D^{m}\left( {- 32} \right)}W^{{- 32}\; n}} +} \\{\sum\limits_{l = {N_{f} + 31}}^{N_{f} - 31}{{D^{m}(l)}W^{{({N_{f} - l})}n}}}\end{pmatrix}}} \\{= {\frac{1}{N_{f}}\left( {W^{{- 32}\; n} + {\sum\limits_{l = {- 31}}^{32}{{D^{m}(l)}W^{{- l}\; n}}}} \right)}} \\{= {d^{m}(n)}}\end{matrix} & \left\lbrack {{Math}.\mspace{14mu} 33} \right\rbrack\end{matrix}$

Therefore, it can be seen that the aforementioned requirements aresatisfied if additional elements D^(m)(−32) and D^(N-m) (−32) have aconjugate relation with each other, that is, D^(m)(−32)=(D^(N-m)(−32))*.

Every function as described above can be performed by a processor suchas a microprocessor based on software coded to perform such function, aprogram code, etc., a controller, a micro-controller, an ASIC(Application Specific Integrated Circuit), or the like. Planning,developing and implementing such codes may be obvious for the skilledperson in the art based on the description of the present invention.

Although the embodiments of the present invention have been disclosedfor illustrative purposes, those skilled in the art will appreciate thatvarious modifications, additions and substitutions are possible, withoutdeparting from the scope of the invention. Accordingly, the embodimentsof the present invention are not limited to the above-describedembodiments but are defined by the claims which follow, along with theirfull scope of equivalents.

1. A method of transmitting a synchronization signal in a wireless communication system, the method comprising: mapping each element of a synchronization signal sequence P(k) to a corresponding subcarrier from a sequence of subcarriers s(n) as shown by: s(n)=P(k) where n=k−31 and k=0, 1, . . . , 30, 32, . . . , 62; and transmitting the mapped synchronization signal sequence P(k), wherein the sequence P(k) is defined by a Zadoff-Chu (ZC) sequence having a length N=63 according to ${P(k)} = {\exp\left\{ {- \frac{j\;\pi\;{{Mk}\left( {k + 1} \right)}}{63}} \right\}}$ where M is a root index which is relatively prime to N and k=0, 1, . . . , 30, 32, . . . , 62, whereby the length of the sequence P(k) is N−1.
 2. The method of claim 1, wherein the synchronization signal is a primary synchronization signal by which a user equipment acquires orthogonal frequency division multiplexing (OFDM) symbol synchronization.
 3. The method of claim 1, wherein the sequence P(k) is not mapped to the DC subcarrier s(0) of the sequence of subcarriers s(n).
 4. The method of claim 1, wherein the synchronization signal is transmitted in the last OFDM symbol in slots 0 and 10 of a radio frame, the radio frame comprising 20 slots, a slot comprising a plurality of OFDM symbols.
 5. The method of claim 1, wherein the root index M represents a cell identity.
 6. A method of acquiring synchronization with a cell in a wireless communication system, the method comprising: receiving a primary synchronization signal; and receiving a secondary synchronization signal, wherein a sequence P(k) for the primary synchronization signal is generated from a ZC sequence having the length N=63 according to ${P(k)} = {\exp\left\{ {- \frac{j\;\pi\;{{Mk}\left( {k + 1} \right)}}{63}} \right\}}$ where M is a root index which is relatively prime to N and k=0, 1, . . . , 30, 32, . . . , 62, and each element of th the sequence P(k) is mapped to a corresponding subcarrier from a sequence of subcarriers s(n) as shown by: s(n)=P(k) where n=k−31 and k=0, 1, . . . , 30, 32, . . . ,
 62. 7. The method of claim 6, wherein the sequence P(k) is not mapped to the DC subcarrier s(0) of the sequence of subcarriers s(n).
 8. The method of claim 6, wherein the primary synchronization signal and the secondary synchronization signal are transmitted in consecutive OFDM symbols.
 9. The method of claim 6, wherein the primary synchronization signal is used to acquire OFDM symbol synchronization and the secondary synchronization signal is used to acquire frame synchronization.
 10. A user equipment for transmitting a synchronization signal in a wireless communication system, the user equipment comprising a processor configured to: mapping each element of a synchronization signal sequence P(k) to a corresponding subcarrier from a sequence of subcarriers s(n) as shown by: s(n)=P(k) where n=k−31 and k=0, 1, . . . , 30, 32, . . . , 62; and transmitting the mapped synchronization signal sequence P(k), wherein the sequence P(k) is defined by a Zadoff-Chu (ZC) sequence having a length N=63 according to ${P(k)} = {\exp\left\{ {- \frac{j\;\pi\;{{Mk}\left( {k + 1} \right)}}{63}} \right\}}$ where M is a root index which is relatively prime to N and k=0, 1, . . . , 30, 32, . . . , 62, whereby the length of the sequence P(k) is N−1. 